Solve the inequality sin ⁡ ( x ) sin ⁡ ( 3 x ) &gt; 1 4 One simple observation is that both x and 3x have to positive or negative simultaneously. I tried expanding sin ⁡ ( 3 x ) by the regular indentity as : sin ⁡ ( x ) × ( 3 sin ⁡ ( x ) − 4 sin 3 ⁡ ( x ) ) &gt; 1 4 ⟹ sin 2 ⁡ ( x ) × ( 3 − 4 sin 2 ⁡ ( x ) ) &gt; 1 4

Question

Solve the inequality   sin  ⁡  (  x  )  sin  ⁡  (  3  x  )  &amp;gt;      1    4  One simple observation is that both x and 3x have to positive or negative simultaneously. I tried expanding   sin  ⁡  (  3  x  ) by the regular indentity as :  sin  ⁡  (  x  )  ×            (        3  sin  ⁡  (  x  )  −  4      sin    3    ⁡  (  x  )            )        &amp;gt;      1    4      ⟹        sin    2    ⁡  (  x  )  ×            (        3  −  4      sin    2    ⁡  (  x  )            )        &amp;gt;      1    4

Lilliana Burton · Accepted Answer

You are on the correct path.Now you need to solve the following equation:      sin    2    ⁡  (  x  )  ⋅            (        3  −  4      sin    2    ⁡  (  x  )            )        =      1    4    4      sin    2    ⁡  (  x  )  ⋅            (        3  −  4      sin    2    ⁡  (  x  )            )        −  1  =  0  12      sin    2    ⁡  x  −  16      sin    4    ⁡  x  −  1  =  0  16      sin    4    ⁡  x  −  12      sin    2    ⁡  x  +  1  =  0So we get by solving,      sin    2    ⁡  x  =            12      ±              144        −        4        ⋅        16                    2      ⋅      16        =            12      ±              80              32    =            6      ±      2              5              16    =            (                                    5                    ±          1                4            )        2    =  (  sin  ⁡      72    ∘        )    2    or  (  sin  ⁡      18    ∘        )    2  So the 4 roots are   x  =      72    ∘  ,  x  =      72    ∘  ,  x  =      18    ∘   and   x  =      18    ∘  .Now observe that, to check the inequality, you have to check for three regions:  x  &amp;lt;      18    ∘        18    ∘    &amp;lt;  x  &amp;lt;      72    ∘    x  &amp;gt;      72    ∘  And see which region(s) satisfy the inequality.

Solve the inequality sin ⁡ ( x ) sin ⁡ ( 3 x

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